Problem: Rewrite as a simplified fraction. ${0.82\overline{7} = {?}}$
Let $x$ equal the decimal. Set up two equations such that the digits after the decimal point are identical. ${\begin{align*}1000x &= 827.7777...\\ 100x &= 82.7777...\end{align*}}$ Subtracting the two equations, we have: ${900x = 745}$ ${x = \dfrac{745}{900}} $ ${x = \dfrac{149}{180}} $